Kuklin Alexey

The work is abstract in nature with elements research activities. It discusses various ways to construct regular n-gons. The work contains a detailed answer to the question of whether it is always possible to construct an n-gon using a compass and a ruler. The work is accompanied by a presentation, which can be found on this mini-site.

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Construction of regular polygons Work completed by: student of grade 9 “B” MBOU secondary school No. 10 Kuklin Alexey

Regular polygons A regular polygon is a convex polygon in which all sides and angles are equal. Go to examples A convex polygon is a polygon all of whose points lie on the same side of any line passing through two of its adjacent vertices.

Back Regular polygons

The founders of the branch of mathematics about regular polygons were ancient Greek scientists. One of them was Archimedes and Euclid.

Proof of the existence of a regular n-gon If n (the number of angles of the polygon) is greater than 2, then such a polygon exists. Let's try to build an 8-gon and prove it. Proof

Let's take a circle of arbitrary radius with a center at point O. Divide it into a certain number of equal arcs, in our case 8. To do this, draw the radii so that we get 8 arcs, and the angle between the two nearest radii is equal to 360°: the number of sides (in our case 8), respectively, each angle will be equal to 45°.

3. We get points A1, A2, A3, A4, A5, A6, A7, A8. We connect them one by one and get a regular octagon. Back

Constructing a regular polygon along a side using rotation A regular polygon can be constructed by knowing its angles. We know that the sum of the angles of a convex n-gon is 180°(n - 2). From this you can calculate the angle of the polygon by dividing the sum by n. Angles Construction

Regular angle: 3-gon is 60° 4-gon is 90° 5-gon is 108° 6-gon is 120° 8-gon is 135° 9-gon is 140° 10-gon is 144° 12-gon is 150 ° Degree measure of angles of regular triangles Back

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In 1796, one of the greatest mathematicians of all time, Carl Friedrich Gauss, showed the possibility of constructing regular n-gons if the equality is satisfied, where n is the number of angles, and k is any natural number. Thus, it turned out that within 30 it is possible to divide the circle into 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30 equal parts. In 1836, Wantzel proved that regular polygons that do not satisfy this equality cannot be constructed using a ruler and compass. Gauss's theorem

Constructing a triangle Let's construct a circle with a center at point O. Let's construct another circle of the same radius passing through point O.

3. Connect the centers of the circles and one of their intersection points, obtaining a regular polygon. Back Constructing a triangle

Construction of a hexagon 1. Construct a circle with center at point O. 2. Conduct straight line through the center of the circle. 3. Draw an arc of a circle of the same radius with the center at the point of intersection of the line with the circle until it intersects with the circle.

4. Draw straight lines through the center of the initial circle and the intersection points of the arc with this circle. 5. We connect the points of intersection of all lines with the original circle and get a regular hexagon. Constructing a hexagon

Construction of a quadrilateral Let's construct a circle with the center at point O. Let's draw 2 mutually perpendicular diameters. From the points at which the diameters touch the circle, draw other circles of a given radius until they intersect (the circles).

Construction of a quadrilateral 4. Draw straight lines through the intersection points of the circles. 5. We connect the points of intersection of the lines and the circle and get a regular quadrilateral.

Constructing an octagon You can construct any regular polygon that has 2 times more angles than the given one. Let's build an octagon using a quadrilateral. Let's connect the opposite vertices of the quadrilateral. Let's draw the bisectors of the angles formed by intersecting diagonals.

4. Connect the points lying on the circle, thereby obtaining a regular octagon. Construction of an octagon

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Construction of a decagon Let's construct a circle with the center at point O. Let's draw 2 mutually perpendicular diameters. Divide the radius of the circle in half and from the resulting point on it draw a circle passing through point O.

Construction of a decagon 4. Draw a segment from the center of the small circle to the point at which the larger circle touches its radius. 5. From the point of contact of the large circle and its radius, draw a circle so that it touches the small one.

Construction of a decagon 6. From the intersection points of the large and the resulting circles, we will draw the circles constructed last time and continue to do so until the adjacent circles touch. 7. Connect the dots and get a decagon.

Construction of a pentagon To build a regular pentagon, when constructing a regular decagon, you need to connect not all points in turn, but through one.

Approximate construction of a regular pentagon using Durer's method Let's construct 2 circles passing through each other's center. Let's connect the centers of a straight line, obtaining one of the sides of the pentagon. Let's connect the intersection points of the circles.

Approximate construction of a regular pentagon using Durer's method 4. Let's draw another circle of the same radius with the center at the intersection point of two other circles. 5. Let's draw 2 segments as shown in the figure.

Approximate construction of a regular pentagon using Durer's method 6. Let us connect the points of contact of these segments with circles with the ends of the constructed side of the pentagon. 7. Let's build it to a pentagon.

Approximate construction of a regular pentagon using the methods of Kovarzyk and Bion

Construction of a regular hexagon inscribed in a circle. The construction of a hexagon is based on the fact that its side is equal to the radius of the circumscribed circle. Therefore, to construct it, it is enough to divide the circle into six equal parts and connect the found points to each other (Fig. 60, a).

A regular hexagon can be built using a straight edge and a 30X60° square. To carry out this construction, we take the horizontal diameter of the circle as the bisector of angles 1 and 4 (Fig. 60, b), construct sides 1 -6, 4-3, 4-5 and 7-2, after which we draw sides 5-6 and 3- 2.

Constructing an equilateral triangle inscribed in a circle. The vertices of such a triangle can be constructed using a compass and a square with angles of 30 and 60° or just one compass.

Let's consider two ways to construct an equilateral triangle inscribed in a circle.

First way(Fig. 61,a) is based on the fact that all three angles of the triangle 7, 2, 3 contain 60°, and the vertical line drawn through point 7 is both the height and the bisector of angle 1. Since the angle is 0-1- 2 is equal to 30°, then to find the side

1-2, it is enough to construct an angle of 30° from point 1 and side 0-1. To do this, install the crossbar and square as shown in the figure, draw line 1-2, which will be one of the sides of the desired triangle. To construct side 2-3, set the crossbar in the position shown by the dashed lines, and draw a straight line through point 2, which will determine the third vertex of the triangle.

Second way is based on the fact that if you build a regular hexagon inscribed in a circle and then connect its vertices through one, you will get an equilateral triangle.

To construct a triangle (Fig. 61, b), mark the vertex-point 1 on the diameter and draw a diametrical line 1-4. Next, from point 4 with a radius equal to D/2, we describe an arc until it intersects with the circle at points 3 and 2. The resulting points will be the other two vertices of the desired triangle.

Constructing a square inscribed in a circle. This construction can be done using a square and a compass.

The first method is based on the fact that the diagonals of the square intersect in the center of the circumscribed circle and are inclined to its axes at an angle of 45°. Based on this, we install the crossbar and square with angles of 45° as shown in Fig. 62, a, and mark points 1 and 3. Next, through these points we draw the horizontal sides of the square 4-1 and 3-2 using a crossbar. Then, using a straight edge, we draw the vertical sides of the square 1-2 and 4-3 along the leg of the square.

The second method is based on the fact that the vertices of the square bisect the arcs of the circle enclosed between the ends of the diameter (Fig. 62, b). We mark points A, B and C at the ends of two mutually perpendicular diameters and from them with a radius y we describe arcs until they intersect each other.

Next, through the intersection points of the arcs we draw auxiliary straight lines, marked in the figure with solid lines. The points of their intersection with the circle will determine vertices 1 and 3; 4 and 2. We connect the vertices of the desired square obtained in this way in series with each other.

Construction of a regular pentagon inscribed in a circle.

To fit a regular pentagon into a circle (Fig. 63), we make the following constructions.

We mark point 1 on the circle and take it as one of the vertices of the pentagon. We divide the segment AO in half. To do this, we describe an arc from point A with the radius AO until it intersects with the circle at points M and B. By connecting these points with a straight line, we get point K, which we then connect to point 1. With a radius equal to the segment A7, we describe an arc from point K until it intersects with the diametrical line AO ​​at point H. By connecting point 1 with point H, we get the side of the pentagon. Then, using a compass solution equal to the segment 1H, describing an arc from vertex 1 to the intersection with the circle, we find vertices 2 and 5. Having made notches from vertices 2 and 5 with the same compass solution, we obtain the remaining vertices 3 and 4. We connect the found points sequentially with each other.

Constructing a regular pentagon along a given side.

To construct a regular pentagon along a given side (Fig. 64), we divide the segment AB into six equal parts. From points A and B with radius AB we describe arcs, the intersection of which will give point K. Through this point and division 3 on line AB we draw a vertical line.

We get point 1-vertex of the pentagon. Then, with a radius equal to AB, from point 1 we describe an arc until it intersects with the arcs previously drawn from points A and B. The intersection points of the arcs determine the pentagon vertices 2 and 5. We connect the found vertices in series with each other.

Construction of a regular heptagon inscribed in a circle.

Let a circle of diameter D be given; you need to fit a regular heptagon into it (Fig. 65). Divide the vertical diameter of the circle into seven equal parts. From point 7 with a radius equal to the diameter of circle D, we describe an arc until it intersects with the continuation of the horizontal diameter at point F. We call point F the pole of the polygon. Taking point VII as one of the vertices of the heptagon, we draw rays from the pole F through even divisions of the vertical diameter, the intersection of which with the circle will determine the vertices VI, V and IV of the heptagon. To obtain vertices / - // - /// from points IV, V and VI, draw horizontal lines until they intersect with the circle. We connect the found vertices sequentially to each other. A heptagon can be constructed by drawing rays from the F pole and through odd divisions of the vertical diameter.

The above method is suitable for constructing regular polygons with any number of sides.

The division of a circle into any number of equal parts can also be done using the data in Table. 2, which provides coefficients that make it possible to determine the dimensions of the sides of regular inscribed polygons.